Filtering Problem for Functionals of Stationary Sequences

TL;DR

This paper addresses optimal linear estimation of functionals of stationary sequences, providing formulas for spectral characteristics under certainty and robust methods under spectral uncertainty.

Contribution

It introduces formulas for spectral characteristics in known spectral density cases and develops minimax-robust estimation methods for uncertain spectral densities.

Findings

Formulas for spectral characteristics with known spectral densities.

Minimax-robust estimation methods for spectral uncertainty.

Identification of least favorable spectral densities.

Abstract

The problem of the mean-square optimal linear estimation of functionals which depend on the unknown values of a stationary stochastic sequence from observations of the sequence with noise is considered. In the case of spectral certainty, where the spectral densities of the sequences are exactly known, we propose formulas for calculating the spectral characteristic and value of the mean-square error of the estimate, which are determined using the Fourier coefficients of some functions from the spectral densities. The minimax-robust method of estimation is applied in the case of spectral uncertainty, where the spectral densities are not exactly known, but a class of admissible spectral densities is given. Formulas for determining the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the functionals are proposed for some specific classes of admissible spectral densities.